OFFSET
0,3
COMMENTS
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 37*x^3 + 2149*x^4 + 532611*x^5 + 539508291*x^6 +...
where
A(x)^2 = 1 + 2*x + 7*x^2 + 80*x^3 + 4381*x^4 + 1069742*x^5 + 1080096067*x^6 +...+ A135756(n)*x^n +...
such that
A135756(n) = Sum_{k=0..n} binomial(n,k) * 2^(k*(k-1)).
The residue of the terms modulo 3 begin:
[1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, ...]
where a(k) appears to be congruent to 1 (mod 3) at k = 9*A005836(n) + [0,1,3,4] for n>=0, and congruent to zeros elsewhere.
PROG
(PARI) {a(n) = polcoeff( sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)*2^(k*(k-1))) +x*O(x^n))^(1/2), n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 25 2015
STATUS
approved